I am not sure what you are asking.
But sometimes I get expressions which look like,

.

And in general there is no way to simplify it, you just leave it like that. Part of the difficulty in the word "simplify" is that it is not well-defined, and hence we cannot come up with a theory which tells us when we can "simplify" and when we cannot. Perhaps you can create a definition.

And, I also do not understand how you got trigonometric functions in your answer. I never remember learning it that.

I am not sure what you are asking.
But sometimes I get expressions which look like,

.

And in general there is no way to simplify it, you just leave it like that. Part of the difficulty in the word "simplify" is that it is not well-defined, and hence we cannot come up with a theory which tells us when we can "simplify" and when we cannot. Perhaps you can create a definition.

And, I also do not understand how you got trigonometric functions in your answer. I never remember learning it that.

The answer is that I would like to see if there is a way I can know that
without resorting to a decimal approximation.

I did make a mistake in my derivation. (That's what I get for not writing it out in full.)

As to how I got that form:

Let and

Then

is one solution.

Now put into complex polar form. (This will "simplify" the eventual answer for x.)